Learn Einstein Notation with Examples and Notes

[Denver Dang]

Hi.

Currently I'm taking an advanced particle physics course, and apparently Einstein notation takes up a lot in this course. Unfortunately for me, and several others in this course, we have never had anything with this kind of notation before. And pretty much from day one, we were put right into it.
I've asked my teacher, but he said: "Read Wikipedia" and solve this problem, which is as hard as it get in this course, and then you should understand Einstein notation. And yes, I probably will if I'm able to solve the problem, but with pretty much no basics regarding Einstein notation, it's really hard getting started, as you could imagine.

So my request here is, if anybody had some notes on Einstein notation, maybe with some problems/examples included, that could really teach me how the Einstein notation works ?
I have tried searching for it online, but I none of what I found was any good in my opinion.

So yes, anybody who can help ?


Thanks in advance.

 

[nrqed]

Hi.

Currently I'm taking an advanced particle physics course, and apparently Einstein notation takes up a lot in this course. Unfortunately for me, and several others in this course, we have never had anything with this kind of notation before. And pretty much from day one, we were put right into it.
I've asked my teacher, but he said: "Read Wikipedia" and solve this problem, which is as hard as it get in this course, and then you should understand Einstein notation. And yes, I probably will if I'm able to solve the problem, but with pretty much no basics regarding Einstein notation, it's really hard getting started, as you could imagine.

So my request here is, if anybody had some notes on Einstein notation, maybe with some problems/examples included, that could really teach me how the Einstein notation works ?
I have tried searching for it online, but I none of what I found was any good in my opinion.

So yes, anybody who can help ?


Thanks in advance.

Do you mean Einstein's summation convention? It is simple: whenever the same index appears upstairs and downstairs, a summation is implied. For example. $A^\mu V_\mu$ stands for

$$\sum_{\mu=0}^3 A^\mu V_\mu = A^0 V_0 + A^1 V_1 + A^2 V_2 + A^3 V_3$$

 

[bolbteppa]

Well, for classical Einstein summation convention (i.e. index gymnastics), Schaums 'Tensor Calculus' & the first chapter of Landau's 'Classical Theory of Fields' develop it classically, it's worth looking in chapters 6 & 8 of Kok's 'Explorations in Mathematical Physics' to attempt to link it to a modern perspective (i.e. using linear algebra). Right now I can't think of a good book that directly illustrates how contravariant vectors are vectors in a vector space while covariant vectors are vectors in it's dual vector space & how you can turn the index notation abbreviations into nice geometric linear algebra ideas, but keep an eye out for this stuff if the above books aren't enough.

 

[Fredrik]

The basics are very easy. For each matrix M, and all i,j such that this makes sense, let $M_{ij}$ denote the number on row i, column j of M. The definition of matrix multiplication is $(AB)_{ij}=\sum_k A_{ik}B_{kj}$. Since the sum is always over the index that appears twice, we don't need the summation sigma to remind us which variable we're summing over, so we simply drop it from the notation and write $(AB)_{ij}=A_{ik}B_{kj}$ instead.

A similar convention is used for the trace of a square matrix: $\operatorname{Tr} A=A_{ii}$. These notations make many proofs look trivial. For example the complete proof of Tr AB = Tr BA looks like this: $(AB)_{ii}=A_{ij}B_{ji} =B_{ji}A_{ij}=(BA)_{jj}$.

This convention is especially nice when we also use the convention to write the component on row i, column j of M as $M^i_j$ rather than $M_{ij}$. In this notation, we have $(AB)^i_j =A^i_k B^k_j$ and $\operatorname{Tr} A=A^i_i$.

A Lorentz transformation can be defined as a linear operator on $\mathbb R^4$ such that (the corresponding matrix satisfies) $\Lambda^T\eta\Lambda=\eta$. Multiply this by $\eta^{-1}$ from the left, and you will see that $\Lambda^{-1}=\eta^{-1}\Lambda^T\eta$. These equalities are often written out in component form. This is a simple application of the definition of matrix multiplication, that's made to look difficult by some notational conventions that seem strange at first. Indices are Greek letters that run from 0 to 3.

The component on row $\mu$, column $\nu$ of $\eta$ is written as $\eta_{\mu\nu}$, and the component on row $\mu$, column $\nu$ of $\eta^{-1}$ is written as $\eta^{\mu\nu}$. For most other matrices M, the convention is $M^\mu{}_\nu$. This can be viewed as the default convention that's used when we don't have a reason to use another.

In addition to these conventions, there's also the convention that $\eta$ and $\eta^{-1}$ are used to raise and lower indices, as illustrated by these examples: $\eta_{\mu\rho}M^\rho{}_\nu =M_{\mu\nu}=M_\mu{}^\rho \eta_{\rho\nu}$.

If we use these conventions, we see that row $\mu$, column $\nu$ of the equality $\eta=\Lambda^T\eta\Lambda$ is
$$\eta_{\mu\nu}=(\Lambda^T\eta\Lambda)_{\mu\nu} =(\Lambda^T)^\mu{}_\rho \eta_{\rho\sigma}\Lambda^\sigma{}_\nu =\Lambda^\rho{}_\mu \eta_{\rho\sigma}\Lambda^\sigma{}_\nu.$$ Similarly, row $\mu$, column $\nu$ of the equality $\Lambda^{-1}=\eta^{-1}\Lambda^T\eta$ is
$$(\Lambda^{-1})^\mu{}_\nu =(\eta^{-1}\Lambda^T\eta)^\mu{}_\nu =\eta^{\mu\rho}(\Lambda^T)^\rho{}_\sigma \eta_{\sigma\nu} =\eta^{\mu\rho}\Lambda^\sigma{}_\rho \eta_{\sigma\nu} =\Lambda_\nu{}^\mu.$$ That last result shows that the horizontal placement of the indices is important, when the metric is used to raise and lower indices as discussed above.

Most books would skip the intermediate steps in these calculation where the index that's summed over appears twice upstairs or twice downstairs, instead of once upstairs and once downstairs.

If you want to understand the conventions for where the indices are placed, you will need to study tensors. My recommendation is chapter 3 in Schutz, "A first course in general relativity". http://books.google.com/books?id=GgRRt7AbdwQC&lpg=PP1&dq=schutz&hl=sv&pg=PA56#v=onepage&q&f=false

 

[SredniVashtar]

Have a look at this old book: "Mathematical Methods in Physics and Engineering" by John W. Dettman. It starts right away with the summation convention. The first five pages might as well be accessible via "look inside".

 

[FactChecker]

Well, for classical Einstein summation convention (i.e. index gymnastics), Schaums 'Tensor Calculus'

I second the recommendation of Schaum's Outlines "Tensor Calculus". If you are not familiar with the Schaum's Outline series, they are always stuffed full of examples and exercises. Great for self-taught drills.

 

[Denver Dang]

Thanks for all the answers. It really helped a lot :)

 

[dextercioby]

Sorry, but how did you get to advanced particle physics without going through special relativity and electrodynamics first ?

 

[Daverz]

Sorry, but how did you get to advanced particle physics without going through special relativity and electrodynamics first ?

I can easily imagine someone never seeing the summation convention and up/down indices if they haven't had a GR or QFT course, or a course that covered the relativity chapters in Jackson (and now I can't remember if Jackson uses the summation convention).

 

[vanhees71]

Jackson uses the summation convention in special relativity, but only very late in the book (Chpt. 11 in the 3rd edition). I never understood, how you can write a (by the way excellent) E+M text as Jackson's and introduce Special Relativity only in the 11th chapter! I'd start at page 1 with an intro to special relativity.

The best book to learn tensor calculus in SRT is Landau/Lifshitz vol. II (Classical Field Theory), which is an excellent treatment of both electromagnetism and gravity (the General Theory of Relativity).